Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | D3×#520 |
| Fitting | C3×#520 ⋊ C2 |
| Sylow | C3 ⋊ C2×#520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting | C3×#520 | C2 |
| socle, cosocle | C22×C3 | C25 |
| prime radical | C2×D3×#41 | C2 |
| hypercenter | #520 | D3 |
| hypocenter | C3 | C2×#520 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #520 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(5,7,8,11)(6,10,9,12)(13,14,15);(6,9)(10,12);(5,6)(7,10)(8,9)(11,12);(1,2,3,4)(6,9)(7,11);(14,15)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1859, D3×#1860, #10796062, … |
| maximal normal subgroup of | D3×#1859, D3×#1860, #10796062, … |
| maximal quotient of | D3×#2274, D3×#2275, D3×#2276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18475);